Chapter 4 of 8

All 8 chapters
  1. 1Coordinates, readunderstood
  2. 2All four quarters of the grid, readunderstood
  3. 3Shapes on a grid, readunderstood
  4. 4From a rule to a line, readunderstood
  5. 5Lines like x = 3 and y = −2, readunderstood
  6. 6Steepness: the gradient, readunderstood
  7. 7The equation y = mx + c, readunderstood
  8. 8Graphs from real life, readunderstood

M3.9.4 Chapter 4

From a rule to a line

A rule gives a y number for every x number. Can I draw the rule as a picture?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Make a table of values for y = 2x + 1
  • Plot the points and join them with a straight line

Here is a rule: y = 2x + 1. It says: double the number x, then add 1. The answer is y. Put in x = 3 and you get y = 2 × 3 + 1 = 7.

Each pair of x and y is a point. For x = 3 and y = 7, the point is (3, 7). A table of values lists several pairs, one in each row.

A table with two columns, x and y = 2x + 1. The rows are −2 and −3, −1 and −1, 0 and 1, 1 and 3, 2 and 5.xy = 2x + 1−2−3−1−1011325
Each time x goes up by 1, y goes up by 2.

Draw it first

Draw the line y = 2x + 1

  1. A grid with the x axis across and the y axis up. Step 1: the point (0, 1), where x is 0. Step 2: all five points from the table: (−2, −3), (−1, −1), (0, 1), (1, 3) and (2, 5). Step 3: a straight line through the points, labelled y = 2x + 1.−5−4−3−2−112345−5−4−3−2123450xy

    1 Start with x = 0. The rule gives y = 1, so plot (0, 1).

  2. A grid with the x axis across and the y axis up. Step 1: the point (0, 1), where x is 0. Step 2: all five points from the table: (−2, −3), (−1, −1), (0, 1), (1, 3) and (2, 5). Step 3: a straight line through the points, labelled y = 2x + 1.−5−4−3−2−112345−5−4−3−2123450xy

    2 Work out y for the other x numbers in the table. Plot each point.

  3. A grid with the x axis across and the y axis up. Step 1: the point (0, 1), where x is 0. Step 2: all five points from the table: (−2, −3), (−1, −1), (0, 1), (1, 3) and (2, 5). Step 3: a straight line through the points, labelled y = 2x + 1.y = 2x + 1−5−4−3−2−112345−5−4−3−2123450xy

    3 Lay a ruler along the points and draw the line. It goes on to the edge of the grid.

Worked example

Make a table of values for y = 3x − 1 using x = −1, 0, 1 and 2.

  1. For x = −1: y = 3 × (−1) − 1 = −3 − 1 = −4.
  2. For x = 0: y = 3 × 0 − 1 = −1.
  3. For x = 1: y = 3 × 1 − 1 = 2.
  4. For x = 2: y = 3 × 2 − 1 = 5.

Answer: −4, −1, 2, 5

A grid with the x axis across and the y axis up. The four points from the table, (−1, −4), (0, −1), (1, 2) and (2, 5), and the straight line y = 3x − 1 through them.y = 3x − 1−5−4−3−2−112345−5−3−2−1123450xy
The four points lie on one straight line.

Worked example

Plot the points of y = 3x − 1 and draw the line. Where does the line cross the y axis?

  1. Plot the four points from the table: (−1, −4), (0, −1), (1, 2) and (2, 5).
  2. Draw a straight line through them.
  3. The line crosses the y axis where x = 0. The table says y = −1 there.
  4. So the line crosses the y axis at (0, −1).

Answer: (0, −1)

The line goes on and on in both directions. Every point on it follows the rule, and every point that follows the rule is on it. A point off the line breaks the rule.

A grid with the x axis across and the y axis up. The line y = x + 1. The dot P (2, 3) is on the line, because 3 = 2 + 1. The open dot Q (3, 2) is below the line, because 2 is not 3 + 1.PQy = x + 1−5−4−3−2−112345−5−4−3−2−1123450xy
P is (2, 3) and Q is (3, 2). Put each point into the rule. P fits it. Q does not.

See it three ways

With things

A seedling is 1 cm tall. It grows 2 cm every day. Measure it each morning.

As a picture

A grid with the x axis across and the y axis up. A graph with days across as x and cm up as y. The points are (0, 1), (1, 3) and (2, 5), on the line y = 2x + 1.y = 2x + 112345612345670dayscm

In symbols

y = 2x + 1, where x is the number of days and y is the height in cm.

Try it

Catch a wrong point

You need: squared paper, a ruler, a pencil.

  1. Draw the axes on squared paper, from 0 to 6 across and 0 to 6 up.
  2. Take the rule y = x + 2. Make a table for x = 0, 1, 2, 3 and 4.
  3. Plot the five points. Lay the ruler along them.
  4. Change one answer in your table on purpose. Plot it and see what happens.

What you will see: The right points all lie on one straight line. A wrong point sits off the line. If a point looks off the line, check that sum again.

If you cannot try it, this is what happens.
A grid with the x axis across and the y axis up. The points (0, 2), (1, 3), (2, 4), (3, 5) and (4, 6) on the line y = x + 2. An open dot W at (3, 6) is off the line.Wy = x + 212345612345670xy

A wrong idea many people have

The line stops at the last point you plotted.

Why it feels right: You only worked out a few points, so it seems the line should end there.

The test that breaks it: Put x = 10 into y = 2x + 1. You get y = 21. That is a point of the rule, far past the points in the table.

What is true: The rule works for every x, so its line goes on in both directions. Plot a few points to find the line, then draw it to the edge of the grid.

Key words

rule
a sentence with x and y, such as y = 2x + 1, that gives y for each x.
table of values
a list of x numbers with the y number that the rule gives for each.

Back to your guess

The answer is A: The points of y = 2x + 1 lie on a straight line.

Each time x goes up by 1, y goes up by 2. Steps that are always the same make a straight line. The line goes on both ways, because the rule works for every x.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Work out y when x = 3 in the rule y = 2x + 1.

    Tap your answer.

    1. Not this one. You doubled 3 but forgot to add 1. Go to M3.9.4. Try another.

    2. Not this one. You added 3 + 2. The 2 in 2x means 2 times x. Go to M3.8.2. Try another.

    3. Yes, that is right. 2 × 3 = 6. Then 6 + 1 = 7.

    4. Not this one. You worked out 2 × 3 = 6 and then added 3. The rule adds 1. Go to M3.9.4. Try another.

    See the answer

    7

    2 × 3 = 6. Then 6 + 1 = 7.

    • If you chose A: You doubled 3 but forgot to add 1. Go to M3.9.4.
    • If you chose B: You added 3 + 2. The 2 in 2x means 2 times x. Go to M3.8.2.
    • If you chose D: You worked out 2 × 3 = 6 and then added 3. The rule adds 1. Go to M3.9.4.

  2. Work out y when x = −1 in the rule y = x − 3.

    Tap your answer.

    1. Not this one. You worked out 1 − 3. But x is −1, so the sum is −1 − 3. Go to M3.1.2. Try another.

    2. Yes, that is right. Put x = −1 into y = x − 3. Then y = −1 − 3 = −4.

    3. Not this one. You added 3 instead of taking it away. The rule says y = x − 3. Go to M3.9.4. Try another.

    4. Not this one. You dropped the minus signs. Both −1 and the −3 matter. Go to M3.1.1. Try another.

    See the answer

    −4

    Put x = −1 into y = x − 3. Then y = −1 − 3 = −4.

    • If you chose A: You worked out 1 − 3. But x is −1, so the sum is −1 − 3. Go to M3.1.2.
    • If you chose C: You added 3 instead of taking it away. The rule says y = x − 3. Go to M3.9.4.
    • If you chose D: You dropped the minus signs. Both −1 and the −3 matter. Go to M3.1.1.

  3. The line y = 2x − 1 is drawn. Which of these points lies on it?

    A grid with the x axis across and the y axis up. The straight line y = 2x − 1 across the grid.y = 2x − 1−5−4−3−2−112345−5−4−3−2−1123450xy
    Tap your answer.

    1. Not this one. Put x = 2 into the rule. You get y = 3, not 5. Go to M3.9.4. Try another.

    2. Not this one. Put x = 1 into the rule. You get y = 1, not 2. Go to M3.9.4. Try another.

    3. Not this one. Put x = 3 into the rule. You get y = 5, not 4. Go to M3.9.4. Try another.

    4. Yes, that is right. Put x = 2 into y = 2x − 1. Then y = 2 × 2 − 1 = 3, so (2, 3) is on the line.

    See the answer

    (2, 3)

    Put x = 2 into y = 2x − 1. Then y = 2 × 2 − 1 = 3, so (2, 3) is on the line.

    • If you chose A: Put x = 2 into the rule. You get y = 3, not 5. Go to M3.9.4.
    • If you chose B: Put x = 1 into the rule. You get y = 1, not 2. Go to M3.9.4.
    • If you chose C: Put x = 3 into the rule. You get y = 5, not 4. Go to M3.9.4.

  4. Tariq made a table for y = 2x + 1. One of his points is wrong. Which one?

    A table with two columns, x and y = 2x + 1. The rows are 0 and 1, 1 and 3, 2 and 5, 3 and 8.xy = 2x + 101132538
    Tap your answer.

    1. Not this one. Check it: 2 × 2 + 1 = 5. This point is right. Go to M3.9.4. Try another.

    2. Yes, that is right. Put x = 3 into the rule: 2 × 3 + 1 = 7, not 8. The point should be (3, 7).

    3. Not this one. Check it: 2 × 0 + 1 = 1. This point is right. Go to M3.9.4. Try another.

    4. Not this one. Check it: 2 × 1 + 1 = 3. This point is right. Go to M3.9.4. Try another.

    See the answer

    (3, 8)

    Put x = 3 into the rule: 2 × 3 + 1 = 7, not 8. The point should be (3, 7).

    • If you chose A: Check it: 2 × 2 + 1 = 5. This point is right. Go to M3.9.4.
    • If you chose C: Check it: 2 × 0 + 1 = 1. This point is right. Go to M3.9.4.
    • If you chose D: Check it: 2 × 1 + 1 = 3. This point is right. Go to M3.9.4.

  5. Where does the line y = 3x − 1 cross the y axis?

    Tap your answer.

    1. Yes, that is right. The line crosses the y axis where x = 0. Then y = 3 × 0 − 1 = −1.

    2. Not this one. On the y axis the across number is 0, so x = 0. Go to M3.9.4. Try another.

    3. Not this one. The 3 is the number in front of x. That is the gradient, not where the line crosses. Go to M3.9.4. Try another.

    4. Not this one. On the y axis the across number is 0, so x = 0. Go to M3.9.4. Try another.

    See the answer

    (0, −1)

    The line crosses the y axis where x = 0. Then y = 3 × 0 − 1 = −1.

    • If you chose B: On the y axis the across number is 0, so x = 0. Go to M3.9.4.
    • If you chose C: The 3 is the number in front of x. That is the gradient, not where the line crosses. Go to M3.9.4.
    • If you chose D: On the y axis the across number is 0, so x = 0. Go to M3.9.4.

  6. Work out y when x = 3 in the rule y = 4x − 2.

    See the answer

    10

    4 × 3 = 12. Then 12 − 2 = 10.

  7. A point on y = x + 5 has x = −2. What is its y number?

    See the answer

    3

    Put x = −2 into y = x + 5. Then y = −2 + 5 = 3.

  8. From an earlier chapter

    What is the midpoint of (0, 0) and (6, 4)?

    See the answer

    (3, 2)

    (0 + 6) ÷ 2 = 3 and (0 + 4) ÷ 2 = 2. The midpoint is (3, 2).

Keep it

Copy this into your exercise book.

A rule gives y for each x. Make a table, plot the points, and draw a straight line through them. The line goes on both ways. A point off the line breaks the rule.

Draw: Draw a grid. Plot (0, 1), (1, 3) and (2, 5). Draw the line y = 2x + 1 through them with a ruler.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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